Area Yield Futures and Futures Options: Risk Management and Hedging

Imagine there exist markets for yield futures contracts as well as ordinary price futures contracts. • Intuitively one would think that a combined use of yield futures contracts and price futures contracts ought to provide a reasonable strategy for insuring revenue. • In the paper this idea is made precise. It is shown that revenue can be secured in by a combined replication of these two contracts. • The relevant dynamic strategy is characterized. It depends only on observable price information in these two separate markets, not on ”unobservables”, like parameters in utility functions of the agents involved. • The identified dynamic strategy is, under certain conditions, equivalent to optimal revenue insurance. Only market risk is considered. • The ability to trade continuously can not be dispensed with. These results should of relevance, since markets for crop yield futures and options have been established. Introduction •Area yield crop insurance, where the index is based on average yield in a given geographical area, has been offered in India, Brazil, Canada and the USA. • Parametric insurance (e.g., rainfall insurance) has been proposed in Canada, India and Mexico. A livestock mortality index has been recently designed to cover herders against livestock losses in Mongolia. • In 1995, the Chicago Board of Trade (CBOT) launched its Crop Yield Insurance (CYI) Futures and Options contracts, but at the moment there is no trade in these contracts. Another example is Nord Pool (The Nordic Power Exchange). • Crop Yield Insurance contracts are designed to provide a hedge for crop yield risk. • In the following we abstract from production costs, and assume zero local price basis (i.e., local cash price equals futures price) and zero yield basis (i.e., individual farm yield equals index yield). • This is to say, we only address market risk, not idiosyncratic risk. We also ignore asymmetric information. • There is a large literature on non-market based risk management and insurance of crop yield, using mostly a one period, expected utility framework. •Yield contracts have been analyzed from the perspective of hedging, using a mean variance approach by Vukina, Li and Holthausen 1996. Minimizing the variance of revenue was the objective in Li and Vukina 1998. The model • Consider two futures markets, one where yield futures options are traded, and one where standard price futures options are traded. • The quantity index q(t) at time t is measured in bushels per acre. • The spot price p(t) at time t is measured in $ per bushel. • The revenue R(t) = q(t)p(t). •A yield futures contract specifies that the payoff of q(T ) bushels per acre t time T , has market price at time t < T given by F q t = E Q t (q(T ) · c). (1) •We could alternatively consider an option on the futures index. (This is outlined in the paper.) • The constant c signifies a conversion factor measured in $ per bushel, so that the futures price is measured in $ per acre. • For example, for the Iowa Corn Yield Insurance Futures (ticker symbol CA) the unit of trading is the Iowa yield estimate times $100 (e.g., a yield of 140.3 bushels per acre gives a contract value of $14, 030). •We set this conversion factor equal to 1 without loss of generality. • Similarly an ordinary futures option contract on price is given by F p t = E Q t (p(T )) measured in $ per bushel. • The linear pricing rule of quantity futures implied by the expression (1) is, of course, far from obvious. • In addition to the usual frictions in ordinary futures markets, like no short sale possibilities of the crop, an additional difficulty arises here, since the index q is not a traded asset. • In Aase 2004 this is resolved by considering the quantity s = pq and identifying s as a spot price process. • Based on the price processes s and p a no-arbitrage model is constructed as permitted by financial theory, where s is identified as the spot price of a leasing contract of agrarian land for the crop in the particular region of consideration. • This solves, at least in theory, the pricing problem of these contracts. • In practice the hedging resulting from this use of the different markets may not be entirely accurate, but then one should perhaps have in mind that the only “perfect hedge” is found in a Japanese garden. • Turning to the dynamics of the two processes p and q, we assume that the process q for quantity and p for price are both defined as follows: dq(t) = μq(t)dt + σq(t)dB(t) (2) • Similarly dp(t) = μp(t)dt + σp(t)dB(t), (3) where B(t) = (B1(t), B2(t)) is a standard two dimensional Brownian motion process. The main result • Consider the product contracts of the form R(t) = p(t)q(t). •We want to investigate whether we can lock in a prespecified “revenue” R(t) at any time t prior to the expiration time T by dynamically trading in the two separate futures options markets described above. • To this end imagine first that a separate market for this type of “revenue” were available. The futures price of this contract we denote by F q·p t , and it must be given as follows under our assumptions: F q·p t = E Q t {q(T ) · p(T )}, 0 ≤ t ≤ T. (4) •Notice that this can be written E Q t {q(T ) · p(T )− F q·p t } = 0, 0 ≤ t ≤ T, (5) the usual starting point for analyzing futures contracts. • Equation (5) implies that if the futures price F q·p t is agreed upon at time t, then no money changes hands when the futures position is initiated. •Recall the main features of a simple futures contract on, say, price. For the holder of one long contract, the payoff at expiration is ∫ T t 1 · dFs = FT − Ft = pT − Ft (6) by the principle of convergence in the futures market, where Ft is the futures price of one contract at time t. • If an agent holds θs futures contracts at time s in the time interval (t, T ], the resettlement gain at time T from this strategy would similarly be ∫ T